Determine how many solutions exist for the system of equations. ${5x-y = -6}$ ${10x-2y = -8}$
Solution: Convert both equations to slope-intercept form: ${5x-y = -6}$ $5x{-5x} - y = -6{-5x}$ $-y = -6-5x$ $y = 6+5x$ ${y = 5x+6}$ ${10x-2y = -8}$ $10x{-10x} - 2y = -8{-10x}$ $-2y = -8-10x$ $y = 4+5x$ ${y = 5x+4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x+6}$ ${y = 5x+4}$ Both equations have the same slope with different y-intercepts. This means the equations are parallel. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ Parallel lines never intersect, thus there are NO SOLUTIONS.